Special Cases With No Solutions
An Apollonius problem is impossible if the given circles are nested, i.e., if one circle is completely enclosed within a particular circle and the remaining circle is completely excluded. This follows because any solution circle would have to cross over the middle circle to move from its tangency to the inner circle to its tangency with the outer circle. This general result has several special cases when the given circles are shrunk to points (zero radius) or expanded to straight lines (infinite radius). For example, the CCL problem has zero solutions if the two circles are on opposite sides of the line since, in that case, any solution circle would have to cross the given line non-tangentially to go from the tangent point of one circle to that of the other.
Read more about this topic: Special Cases Of Apollonius' Problem
Famous quotes containing the words special, cases and/or solutions:
“The great rule: If the little bit you have is nothing special in itself, at least find a way of saying it that is a little bit special.”
—G.C. (Georg Christoph)
“Only by being guilty of Folly does mortal man in many cases arrive at the perception of Sense. A thought which should forever free us from hasty imprecations upon our ever-recurring intervals of Folly; since though Folly be our teacher, Sense is the lesson she teaches; since, if Folly wholly depart from us, Further Sense will be her companion in the flight, and we will be left standing midway in wisdom.”
—Herman Melville (18191891)
“Those great ideas which come to you in your sleep just before you awake in morning, those solutions to the worlds problems which, in the light of day, turn out to be duds of the puniest order, couldnt they be put to some use, after all?”
—Robert Benchley (18891945)